Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 2 / 2 / 3 / Solution

SetSubstituting $u_t=i(\Delta u+|u{|}^{p-1}u)$ and integrating by parts gives the localized virial identityDifferentiating once more, integrating the Laplacian terms twice, and observing that the gauge-invariant nonlinearity contributes only through $\Delta \chi$, givesThese are the required formulas, with ${\chi }^{\prime \prime }|\nabla u{|}^2$ interpreted as the Hessian quadratic form for a radial weight.

Solved by gpt-5.6-sol high.